Can you derive a general form (not keeping either volume or pressure constant) for heat capacity from the first law of thermodynamics? Do you have to make any assumptions to do so?
It sounds trival, but I can't seem to work something out:
$$dU = \delta Q - PdV$$
$$dU = \left( \frac{\partial U}{\partial T} \right) dT + \left( \frac{\partial U}{\partial V} \right) dV + \left( \frac{\partial U}{\partial P} \right) dP$$
$$dV = \left( \frac{\partial V}{\partial T} \right)_P dT + \left( \frac{\partial V}{\partial P} \right)_T dP$$
$$\delta Q = \left( \left( \frac{\partial U}{\partial T} \right) + P \left( \frac{\partial V}{\partial T} \right) \right) dT + \left( \left( \frac{\partial U}{\partial P} \right) + P \left( \frac{\partial V}{\partial P} \right) \right) dP + \left( \frac{\partial U}{\partial T}\right)dV$$
But from there, don't you end up having to hold P and V constant to get $\frac{\partial Q}{\partial T}$ (the heat capacity)?
No comments:
Post a Comment